GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 39

Question

Consider a matrix $\left[\begin{array}{lll}0 & 1 & 0 \\ a & 2 & d \\ b & 3 & c\end{array}\right]$. The matrix cannot have rank.

• A. $0$
• B. $1$
• C. $2$
• D. $3$

   

Comments

$0$ rank possible only for $Zero\; matrix$
option $A$

Answer 1

• Rank of a matrix: The rank of a matrix represents the maximum number of linearly independent rows or columns in the matrix.
• Zero rank: A matrix has rank 0 only if all its elements are zero.

In the given matrix, even when a = b = c = 0, the second row still has a non-zero element (2). Therefore, it cannot have rank 0, as it has at least one linearly independent row.

 

• Rank 1: If all rows are linearly dependent (e.g., a = 0, b = 1, c = 2), the matrix will have rank 1.
• Rank 2: If there are two linearly independent rows (e.g., a = 1, b = 0, c = 1), the matrix will have rank 2.
• Rank 3: If all three rows are linearly independent (e.g., a = 1, b = 2, c = 3), the matrix will have rank 3.

Answer 2

using row operations to get

$$\begin{array} {|r|r|}\hline 0 & 1 & 0 \\ \hline a & 0 & d \\ \hline b & 0 & c \\ \hline  \end{array}$$

1. now put a=b=c=d=0
   we get rank=1 $$\begin{array} {|r|r|}\hline 0 & 1 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline  \end{array}$$
2. put a=d=c=0 and b $\ne$ 0
   we get rank=2 $$\begin{array} {|r|r|}\hline 0 & 1 & 0 \\ \hline 0 & 0 & 0 \\ \hline b & 0 & 0 \\ \hline  \end{array}$$
3. put b = 0 and d =0  and a $\ne$ 0 and c $\ne$ 0
   we get rank=3 $$\begin{array} {|r|r|}\hline 0 & 1 & 0 \\ \hline a & 0 & 0 \\ \hline 0 & 0 & c \\ \hline  \end{array}$$

4. proving rank is always atleast 1
   converting original matrix to row echelon form
   r2 = r2 – 2r1
   r3 = r3 – 3r1
   r3 = r3 – (b/a)r2
   r2 = r2 – $\frac{d}{c – (bd)/a}$r3
   switch r1 and r2

   
   $$\begin{array} {|r|r|}\hline a & 0 & 0 \\ \hline 0 & 1 & 0\\ \hline 0 & 0 & c - (bd)/a \\ \hline  \end{array}$$

   as you can see row 2 will always persist no matter what so rank is always at least 1

   so ANS – A

Comments

Why rank can not be 3? What if this is the matrix?

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you are right i messed up in my calculations , i have edited my answer

Answer 3

Ans isA( 0) , at any value of a,b,c,d , the value of rank of matrix can't be zero .